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Variational analysis of spectral functions simplified

Dmitriy Drusvyatskiy(ddrusv***at***uw.edu)
Courtney Kempton(yumiko88***at***uw.edu)

Abstract: Spectral functions of symmetric matrices -- those depending on matrices only through their eigenvalues -- appear often in optimization. A cornerstone variational analytic tool for studying such functions is a formula relating their subdifferentials to the subdifferentials of their diagonal restrictions. This paper presents a new, short, and revealing derivation of this result. We then round off the paper with an illuminating derivation of the second derivative of twice differentiable spectral functions, highlighting the underlying geometry. All of our arguments have direct analogues for spectral functions of Hermitian matrices, and for singular value functions of rectangular matrices.

Keywords: Eigenvalues, singular values, nonsmooth analysis, proximal mapping, subdifferential, Hessian, group actions.

Category 1: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation: arXiv:1506.05170v2

Download: [PDF]

Entry Submitted: 07/27/2015
Entry Accepted: 07/27/2015
Entry Last Modified: 07/27/2015

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